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  1. MHB Apprentice

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    #1
    I'm confused on this question.

    The equation m log p (n) = q can be written in exponential form as..
    The answer on the work sheet is p^(q/m)=n but shouldn't it be P^(qm) = n ? According to the power rule? My teacher explained this by writing down for me log p (n) = q / m but I'm confused here

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    #2
    Quote Originally Posted by zekea View Post
    I'm confused on this question.

    The equation m log p (n) = q can be written in exponential form as..
    The answer on the work sheet is p^(q/m)=n but shouldn't it be P^(qm) = n ? According to the power rule? My teacher explained this by writing down for me log p (n) = q / m but I'm confused here
    The definition of the logarithm function id the following:
    If $b$ is any number such that $b>0$ and $b\neq 1$ and $x>0$ then,
    $$y=\log_b x \ \ \text{ is equivalent to } \ \ b^y=x$$


    We have the the equation $q=m\log_p n$.

    Dividing both sides by $m$ we get $$\frac{q}{m}=\frac{m\log_p n}{m} \Rightarrow \frac{q}{m}=\log_p n$$

    Therefore from the definition for $y=\frac{q}{m}$, $b=p$ and $x=n$ we get $$ p^{\frac{q}{m}}=n$$
    Last edited by mathmari; December 15th, 2016 at 21:23.

  3. MHB Apprentice

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    #3 Thread Author
    Okay this is what my teacher did but something is confusing me.

    Based on Khans' video here
    Skip to 4:00
    Basically according to the power rule you have Log a (c)^d = bd . He brought the d down to the other side.
    So in exp form A^(bd) = C^d so shouldn't the answer be p^(mn) = n rather than P^(q/m) = n ?

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    #4
    Using that rule we have the following:
    $$q=m\log_p n\Rightarrow q=\log_p n^m$$

    Then from the definition we get $p^q=n^m$.

    To solve for $n$ we do the following: $$n^m=p^q \Rightarrow \left (n^m\right )^{\frac{1}{m}}=\left (p^q\right )^{\frac{1}{m}} \Rightarrow n^{\frac{m}{m}}=p^{\frac{q}{m}} \Rightarrow n=p^{\frac{q}{m}}$$

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    #5
    Quote Originally Posted by zekea View Post
    Okay this is what my teacher did but something is confusing me.

    Based on Khans' video here
    Skip to 4:00
    Basically according to the power rule you have Log a (c)^d = bd . He brought the d down to the other side.
    So in exp form A^(bd) = C^d so shouldn't the answer be p^(mn) = n rather than P^(q/m) = n ?
    If I was given:

    $ \displaystyle \log_a\left(c^d\right)=bd$

    I would first use the identity $\log_a\left(b^c\right)=c\cdot\log_a(b)$ to write:

    $ \displaystyle d\cdot\log_a\left(c\right)=bd$

    Next, divide through by $d$:

    $ \displaystyle \log_a\left(c\right)=b$

    Finally, convert from logarithmic to exponential form:

    $ \displaystyle c=a^b$

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